Question: Solve for $x$ : $ 2|x - 3| - 1 = 5|x - 3| + 5 $
Solution: Subtract $ {2|x - 3|} $ from both sides: $ \begin{eqnarray} 2|x - 3| - 1 &=& 5|x - 3| + 5 \\ \\ {- 2|x - 3|} && {- 2|x - 3|} \\ \\ -1 &=& 3|x - 3| + 5 \end{eqnarray} $ Subtract $5$ from both sides: $ \begin{eqnarray} -1 &=& 3|x - 3| + 5 \\ \\ {- 5} && {- 5} \\ \\ -6 &=& 3|x - 3| \end{eqnarray} $ Divide both sides by ${3}$ $ \dfrac{-6} {{3}} = \dfrac{3|x - 3|} {{3}} $ Simplify: $ -2 = |x - 3| $ The absolute value cannot be negative. Therefore, there is no solution.